Question

Exhaust gases from the turbine of a simple Brayton cycle are quite hot and may be used for other thermal purposes. One proposed use is generating saturated steam at \(110^{\circ} \mathrm{C}\) from water at \(30^{\circ} \mathrm{C}\) in a boiler. This steam will be distributed to several buildings on a college campus for space heating. A Brayton cycle with a pressure ratio of 6 is to be used for this purpose. Plot the power produced, the flow rate of produced steam, and the maximum cycle temperature as functions of the rate at which heat is added to the cycle. The temperature at the turbine inlet is not to exceed \(2000^{\circ} \mathrm{C}\)

Short answer

Answer: In a simple Brayton cycle, the relationships between the power produced, the flow rate of steam, and the maximum cycle temperature concerning the heat added can be represented by the following plots: 1. Plot of net work output (W_net) as a function of heat addition (Q) 2. Plot of mass flow rate of produced steam (m_steam) as a function of heat addition (Q) 3. Plot of maximum cycle temperature (T3) as a function of heat addition (Q) These relationships are derived from thermodynamic equations specific to the Brayton cycle, assuming that all the exhaust heat is used to heat the water to saturated steam.

Step-by-step solution

Identify the Brayton cycle parameters

First, identify all the key parameters: - Pressure ratio (rp): 6 - Turbine inlet temperature (TIT): not to exceed \(2000^{\circ} \mathrm{C}\) - Temperature at which steam is generated: \(110^{\circ} \mathrm{C}\) - Water inlet temperature: \(30^{\circ} \mathrm{C}\)

Calculate the specific heat ratios

For an ideal gas in a simple Brayton cycle, we can assume constant specific heat ratios (Cp and Cv). Since the working fluid is air, we can assign the following values: - Specific heat at constant pressure (Cp): 1.005 kJ/kgK - Specific heat at constant volume (Cv): 0.718 kJ/kgK - Gamma (γ) = Cp / Cv Calculate γ: γ = 1.005 / 0.718 = 1.4

Calculate the temperature rise across the compressor and the turbine

Using the pressure ratio and γ, we can calculate the temperature rise (ΔT) across the compressor and the turbine. For the compressor (from state 1 to state 2): - ΔT_compressor = T2 - T1 = T1 * ((rp)^((γ - 1)/γ) - 1) For the turbine (from state 3 to state 4): - ΔT_turbine = T3 - T4 = T3 * (1 - (rp)^((-γ + 1)/γ))

Calculate the net work output and the heat addition

The net work output (W_net) of the cycle is the difference between the work done by the turbine and the work required by the compressor: W_net = W_turbine - W_compressor = m(T3 - T4) - m(T2 - T1) The heat addition can be calculated by finding the temperature rise at the heat exchanger from state 2 to state 3: Q = m * Cp * (T3 - T2)

Calculate the flow rate of produced steam

Assuming that all the exhaust heat is used to heat the water to saturated steam, the mass flow rate of the produced steam (m_steam) can be calculated. The heat transfer from the cycle to the water and steam can be determined by: m_steam * Cp_steam * (T_sat - T_water) = Q Where: - Cp_steam: Specific heat capacity of steam, approximately 4.18 kJ/kgK - T_sat: Temperature of saturated steam, \(110^{\circ} \mathrm{C}\) - T_water: Water inlet temperature, \(30^{\circ} \mathrm{C}\)

Plot the required plots

Using the relationships derived in Steps 3, 4, and 5, the following plots can be made: 1. Plot W_net as a function of Q 2. Plot m_steam as a function of Q 3. Plot the maximum cycle temperature (T3) as a function of Q Ensure that the turbine inlet temperature does not exceed the given limit of \(2000^{\circ} \mathrm{C}\) for any values of Q. Once all these calculations are done, the resulting plots will help visualize the relationship between power produced, flow rate of steam, and maximum cycle temperature concerning the heat added to the Brayton cycle.